A377291 -id:A377291 - cp's OEIS frontend

A374602 Array of successive integer solutions to sqrt((d-c)b^2 + c(b+1)^2) for nonsquare integers d >= 2 (d=A000037(n) for n >= 1), where b and c are positive integers and c < d, read by antidiagonals.

5, 29, 3, 169, 11, 5, 985, 41, 13, 3, 5741, 153, 34, 7, 4, 33461, 571, 89, 18, 5, 10, 195025, 2131, 233, 29, 11, 11, 4, 1136689, 7953, 610, 69, 28, 23, 5, 7, 6625109, 29681, 1597, 178, 62, 58, 13, 8, 6, 38613965, 110771, 4181, 287, 79, 338, 14, 13, 22, 4

Offset: 1

Charles L. Hohn, Jul 13 2024

T(n,k) is the diagonal lengths of increasingly nearly regular d-dimensional Pythagorean hyperrectangles.
Each row n divides into equal length, geometrically periodic subsequences, each with its own subsequence period length (A377290) and geometric growth factor (A377291); it is conjectured that this is the case for all n, and that all solutions conform as such and that there are no solutions that do not, but these are not proven.
It is also not known if there is an algorithm for generating values for all rows other than testing all possible values for a row until a subsequence pattern emerges.
Square d produce solutions following a different pattern, shown as A375336.

			n=row index; d=nonsquare integer of index n (A000037(n)):
 n    d   T(n,k)
---+----+-------------------------------------------------------------
 1 |  2 |  5, 29, 169, 985, 5741, 33461, 195025, 1136689, 6625109, ...
 2 |  3 |  3, 11,  41, 153,  571,  2131,   7953,   29681,  110771, ...
 3 |  5 |  5, 13,  34,  89,  233,   610,   1597,    4181,   10946, ...
 4 |  6 |  3,  7,  18,  29,   69,   178,    287,     683,    1762, ...
 5 |  7 |  4,  5,  11,  28,   62,    79,    175,     446,     988, ...
 6 |  8 | 10, 11,  23,  58,  338,   373,    781,    1970,   11482, ...
 7 | 10 |  4,  5,  13,  14,   25,    62,    111,     148,     185, ...
 8 | 11 |  7,  8,  13,  32,   57,   139,    158,     259,     638, ...
 9 | 12 |  6, 22,  39,  69,   82,   125,    306,     543,    1142, ...
10 | 13 |  4,  5,   7,  17,   30,    43,     53,      76,     185, ...
11 | 14 |  9, 11,  14,  19,   46,    81,    267,     329,     418, ...
12 | 15 |  6, 10,  21,  23,   30,    39,     94,     165,     362, ...
13 | 17 | 25, 27,  34,  41,   98,   171,    260,    1649,    1779, ...
14 | 18 |  6, 13,  15,  18,   21,    50,     87,     132,     198, ...
15 | 19 |  5,  7,   8,   9,   11,    31,     34,      37,      56, ...
16 | 20 | 10, 26,  68, 125,  159,   178,    197,     466,     807, ...
17 | 21 |  6,  9,  12,  13,   14,    33,     57,      86,     134, ...
18 | 22 |  5,  7,   8,  17,   18,    19,     31,      64,      77, ...
19 | 23 | 16, 19,  27,  28,   29,    68,    117,     176,     764, ...
20 | 24 |  6,  9,  11,  14,   36,    39,     57,      58,      59, ...
...
sqrt((2-1)*1^2 + 1*(1+1)^2) = sqrt(5) -> not an integer so not included.
sqrt((2-1)*3^2 + 1*(3+1)^2) = 5 -> T(1,1).
sqrt((2-1)*20^2 + 1*(20+1)^2) = 29 -> T(1,2).
sqrt((3-2)*1^2 + 2*(1+1)^2) = 3 -> T(2,1).
sqrt((6-2)*7^2 + 2*(7+1)^2) = 18 -> T(4,3).

Row 1 is A001653 starting at n=2.
Row 2 is A079935 starting at n=2.
Bisection of row 2 starting with the first term is A189356 starting at n=1.
Bisection of row 2 starting with the second term is A122769 starting at n=2.
Row 3 is A001519 starting at n=3.
Bisection of row 3 starting with the first term is A033889 starting at n=1.
Bisection of row 3 starting with the second term is A033891 starting at n=1.
Row 4 is A131093 starting at n=3.
Cf. A000037, A373666.
Cf. A377290, A377291, A375336.

row(n, c)=my(v=List(), d=n+floor(sqrt(n)+1/2) /* d=A000037(n) */, t=ceil(sqrt(d))); while(#v

T(n, 1) = A373666(A000037(n)).

A377290 For each row n in array A374602(n, k), the period size, as a count of terms, that divides the row into congruent subsequences.

Original entry on oeis.org

1, 2, 2, 6, 4, 4, 14, 10, 8, 10, 6, 8, 14, 16, 34, 10, 8, 34, 8, 12, 22, 22, 32, 18, 18, 30, 14, 18, 16, 12, 38, 22, 28, 26, 42, 20, 74, 36, 14, 54, 12, 16, 34, 38, 54, 26, 58, 50, 24, 36, 102, 46, 32, 78, 14, 22, 38, 46, 118, 22, 30, 68, 36, 32, 130, 74, 34

Offset: 1

Charles L. Hohn, Oct 23 2024

nonn

Here "congruent" means: 1) In the defining formula of A374602: sqrt((d-c)*b^2 + c*(b+1)^2), A374602(n, k) and A374602(n, k+a(n)) have equal c values (see Example), and also 2) A374602(n, k+a(n))/A374602(n, k) converges to a limit as k->oo, shown in A377291.

			Given formula sqrt((d-c)*b^2 + c*(b+1)^2) from A374602, for n=5, the first few terms of A374602(5, k) are:
sqrt((7-3)*1^2 + 3*(1+1)^2) = 4,
sqrt((7-6)*1^2 + 6*(1+1)^2) = 5,
sqrt((7-1)*4^2 + 1*(4+1)^2) = 11,
sqrt((7-4)*10^2 + 4*(10+1)^2) = 28,
sqrt((7-3)*23^2 + 3*(23+1)^2) = 62,
sqrt((7-6)*29^2 + 6*(29+1)^2) = 79,
sqrt((7-1)*66^2 + 1*(66+1)^2) = 175,
sqrt((7-4)*168^2 + 4*(168+1)^2) = 446,
producing the repeating pattern of c values {3, 6, 1, 4}, of length 4 -> a(5).

Charles L. Hohn, Table of n, a(n) for n = 1..90

Cf. A374602, A377291.

cp's OEIS Frontend

A374602 Array of successive integer solutions to sqrt((d-c)b^2 + c(b+1)^2) for nonsquare integers d >= 2 (d=A000037(n) for n >= 1), where b and c are positive integers and c < d, read by antidiagonals.

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A377290 For each row n in array A374602(n, k), the period size, as a count of terms, that divides the row into congruent subsequences.

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cp's OEIS Frontend

A374602 Array of successive integer solutions to sqrt((d-c)*b^2 + c*(b+1)^2) for nonsquare integers d >= 2 (d=A000037(n) for n >= 1), where b and c are positive integers and c < d, read by antidiagonals.

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Author

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Comments

Examples

Crossrefs

Programs

PARI

Formula

A377290 For each row n in array A374602(n, k), the period size, as a count of terms, that divides the row into congruent subsequences.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A374602 Array of successive integer solutions to sqrt((d-c)b^2 + c(b+1)^2) for nonsquare integers d >= 2 (d=A000037(n) for n >= 1), where b and c are positive integers and c < d, read by antidiagonals.